Linear Dual Algebra Algorithms and their Application to Kinematics

Pennestrı̀, Ettore; Valentini, Pier Paolo

doi:10.1007/978-1-4020-8829-2_11

Cited by 40 publications

(37 citation statements)

References 15 publications

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“…One of the most important non-vectorial parameterizations for the orthogonal dual tensor SO 3 is given by the dual quaternions [20,21]. A dual quaternion can be defined as an associated pair of a dual scalar quantity and a free dual vector:…”

Section: Dual Quaternion Parameterizationmentioning

confidence: 99%

“…Orthogonal dual tensorial maps are a powerful instrument in the study of the rigid motion with respect to an inertial and noninertial reference frames. More on dual numbers, dual vectors and dual tensors can be found in [2,[16][17][18][19][20][21][22][23].2.1. Isomorphism between Lie group of the rigid displacements SE 3 and Lie group of the orthogonal dual tensors SO 3…”

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confidence: 99%

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On Six DOF Relative Orbital Motion of Satellites

Condurache¹

2018

Space Flight

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In this chapter, we reveal a dual-tensor-based procedure to obtain exact expressions for the six degree of freedom (6-DOF) relative orbital law of motion in the specific case of two Keplerian confocal orbits. The result is achieved by pure analytical methods in the general case of any leader and deputy motion, without singularities or implying any secular terms. Orthogonal dual tensors play a very important role, with the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the 6-DOF orbital motion problem. The solution does not depend on the local-vertical-local-horizontal (LVLH) properties involves that is true in any reference frame of the leader with the origin in its mass center. A representation theorem is provided for the full-body initial value problem. Furthermore, the representation theorems for rotation part and translation part of the relative motion are obtained.Keywords: relative orbital motion, full body problem, dual algebra, Lie group, Lie algebra, closed form solution IntroductionThe relative motion between the leader and the deputy in the relative motion is a six-degreesof-freedom (6-DOF) motion engendered by the joining of the relative translational motion with the rotational one. Recently, the modeling of the 6-DOF motion of spacecraft gained a special attention [1][2][3][4][5], similar to the controlling the relative pose of satellite formation that became a very important research subject [6][7][8][9][10]. The approach implies to consider the relative translational and rotational dynamics in the case of chief-deputy spacecraft formation to be modeled using vector and tensor formalism.In this chapter we reveal a dual algebra tensor based procedure to obtain exact expressions for the six D.O.F relative orbital law of motion for the case of two Keplerian confocal orbits.© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Orthogonal dual tensors play a very important role, the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the six D.O.F relative orbital motion problem. Because the solution does not depend on the LVLH properties involves that is true in any reference frame of the Leader with the origin in its mass center. To obtain this solution, one has to know only the inertial motion of the Leader spacecraft and the initial conditions of the deputy satellite in the local-vertical-local-horizontal (LVLH) frame. For the full body initial value problem, a general representation theorem is given. More, the real and imaginary parts are split and representation theorems for the rotation and translation parts of the relative orbital motion are obtained. Regarding translation, we w...

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Section: Dual Quaternion Parameterizationmentioning

confidence: 99%

mentioning

confidence: 99%

On Six DOF Relative Orbital Motion of Satellites

Condurache¹

2018

Space Flight

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“…Given Q by means of [16], [9], [10] one can extract d, e 0 , e 1 , e 2 and e 3 . Thus the transform matrix …”

Section: Coordinate Tranform By Means Of Dual Quaternionsmentioning

confidence: 99%

“…Dual algebra can be conveniently used to retrieve the screw parameters of a rigid body motion from the measurement of point coordinates [9,10].…”

Section: How To Obtain Screw Parameters From Point Coordinatesmentioning

confidence: 99%

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Dual Quaternions as a Tool for Rigid Body Motion Analysis: A Tutorial with an Application to Biomechanics

Pennestrı̀¹,

Valentini²

2010

Archive of Mechanical Engineering

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Dual quaternions and dual quaternion interpolation are powerful mathematical tools for the spatial analysis of rigid body motions. In this paper, after a review of some basic results and formulas, it will be presented an attempt to use these tools for the the kinematic modeling of human joints. In particular, the kinematic parameters extracted from experimentally acquired data are compared with those theoretically computed from dual quaternions rigid body motion interpolation.

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Higher-Order Kinematics of Rigid Bodies. A Tensors Algebra Approach

Condurache

2019

Mechanisms and Machine Science

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Linear Dual Algebra Algorithms and their Application to Kinematics

Cited by 40 publications

References 15 publications

On Six DOF Relative Orbital Motion of Satellites

On Six DOF Relative Orbital Motion of Satellites

Dual Quaternions as a Tool for Rigid Body Motion Analysis: A Tutorial with an Application to Biomechanics

Higher-Order Kinematics of Rigid Bodies. A Tensors Algebra Approach

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